This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon. [1] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen. | A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image. |
In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real dimensions, , while a complex polygon exists in two complex dimensions, , which can be given real representations in 4 dimensions, , which then must be projected down to 2 or 3 real dimensions to be visualized. A complex polygon is generalized as a complex polytope in .
A complex polygon may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.
The regular complex polygons have been completely characterized, and can be described using a symbolic notation developed by Coxeter.
A regular complex polygon with all 2-edges can be represented by a graph, while forms with k-edges can only be related by hypergraphs. A k-edge can be seen as a set of vertices, with no order implied. They may be drawn with pairwise 2-edges, but this is not structurally accurate.
While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons p{4}2, are limited to 5-edge (pentagonal edges) elements, and infinite regular apeirogons also include 6-edge (hexagonal edges) elements.
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
A more modern notation p1{q}p2 is due to Coxeter, [2] and is based on group theory. As a symmetry group, its symbol is p1[q]p2.
The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q−1)/2R2 = (R1R2)(q−1)/2R1. When q is odd, p1=p2.
For 4[4]2 has R14 = R22 = I, (R2R1)2 = (R1R2)2.
For 3[5]3 has R13 = R23 = I, (R2R1)2R2 = (R1R2)2R1.
Coxeter also generalised the use of Coxeter–Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram . The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or .
One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So and are ordinary, while is starry.
12 irreducible Shephard groups with their subgroup index relations. [3] | Subgroups from <5,3,2>30, <4,3,2>12 and <3,3,2>6 |
Subgroups relate by removing one reflection: p[2q]2 --> p[q]p, index 2 and p[4]q --> p[q]p, index q. |
Coxeter enumerated this list of regular complex polygons in . A regular complex polygon, p{q}r or , has p-edges, and r-gonal vertex figures. p{q}r is a finite polytope if (p + r)q > pr(q − 2).
Its symmetry is written as p[q]r, called a Shephard group , analogous to a Coxeter group, while also allowing unitary reflections.
For nonstarry groups, the order of the group p[q]r can be computed as . [4]
The Coxeter number for p[q]r is , so the group order can also be computed as . A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.
The rank 2 solutions that generate complex polygons are:
Group | G3 = G(q,1,1) | G2 = G(p,1,2) | G4 | G6 | G5 | G8 | G14 | G9 | G10 | G20 | G16 | G21 | G17 | G18 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2[q]2, q = 3,4... | p[4]2, p = 2,3... | 3[3]3 | 3[6]2 | 3[4]3 | 4[3]4 | 3[8]2 | 4[6]2 | 4[4]3 | 3[5]3 | 5[3]5 | 3[10]2 | 5[6]2 | 5[4]3 | |
Order | 2q | 2p2 | 24 | 48 | 72 | 96 | 144 | 192 | 288 | 360 | 600 | 720 | 1200 | 1800 |
h | q | 2p | 6 | 12 | 24 | 30 | 60 |
Excluded solutions with odd q and unequal p and r are: 6[3]2, 6[3]3, 9[3]3, 12[3]3, ..., 5[5]2, 6[5]2, 8[5]2, 9[5]2, 4[7]2, 9[5]2, 3[9]2, and 3[11]2.
Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .
The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors. [5]
The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.
The group p[q]r, , can be represented by two matrices: [6]
Name | R1 | R2 |
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Order | p | r |
Matrix |
With
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Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes. [7]
Group | Order | Coxeter number | Polygon | Vertices | Edges | Notes | ||
---|---|---|---|---|---|---|---|---|
G(q,q,2) 2[q]2 = [q] q = 2,3,4,... | 2q | q | 2{q}2 | q | q | {} | Real regular polygons Same as Same as if q even |
Group | Order | Coxeter number | Polygon | Vertices | Edges | Notes | |||
---|---|---|---|---|---|---|---|---|---|
G(p,1,2) p[4]2 p=2,3,4,... | 2p2 | 2p | p(2p2)2 | p{4}2 | | p2 | 2p | p{} | same as p{}×p{} or representation as p-p duoprism |
2(2p2)p | 2{4}p | 2p | p2 | {} | representation as p-p duopyramid | ||||
G(2,1,2) 2[4]2 = [4] | 8 | 4 | 2{4}2 = {4} | 4 | 4 | {} | same as {}×{} or Real square | ||
G(3,1,2) 3[4]2 | 18 | 6 | 6(18)2 | 3{4}2 | 9 | 6 | 3{} | same as 3{}×3{} or representation as 3-3 duoprism | |
2(18)3 | 2{4}3 | 6 | 9 | {} | representation as 3-3 duopyramid | ||||
G(4,1,2) 4[4]2 | 32 | 8 | 8(32)2 | 4{4}2 | 16 | 8 | 4{} | same as 4{}×4{} or representation as 4-4 duoprism or {4,3,3} | |
2(32)4 | 2{4}4 | 8 | 16 | {} | representation as 4-4 duopyramid or {3,3,4} | ||||
G(5,1,2) 5[4]2 | 50 | 25 | 5(50)2 | 5{4}2 | 25 | 10 | 5{} | same as 5{}×5{} or representation as 5-5 duoprism | |
2(50)5 | 2{4}5 | 10 | 25 | {} | representation as 5-5 duopyramid | ||||
G(6,1,2) 6[4]2 | 72 | 36 | 6(72)2 | 6{4}2 | 36 | 12 | 6{} | same as 6{}×6{} or representation as 6-6 duoprism | |
2(72)6 | 2{4}6 | 12 | 36 | {} | representation as 6-6 duopyramid | ||||
G4=G(1,1,2) 3[3]3 <2,3,3> | 24 | 6 | 3(24)3 | 3{3}3 | 8 | 8 | 3{} | Möbius–Kantor configuration self-dual, same as representation as {3,3,4} | |
G6 3[6]2 | 48 | 12 | 3(48)2 | 3{6}2 | 24 | 16 | 3{} | same as | |
3{3}2 | starry polygon | ||||||||
2(48)3 | 2{6}3 | 16 | 24 | {} | |||||
2{3}3 | starry polygon | ||||||||
G5 3[4]3 | 72 | 12 | 3(72)3 | 3{4}3 | 24 | 24 | 3{} | self-dual, same as representation as {3,4,3} | |
G8 4[3]4 | 96 | 12 | 4(96)4 | 4{3}4 | 24 | 24 | 4{} | self-dual, same as representation as {3,4,3} | |
G14 3[8]2 | 144 | 24 | 3(144)2 | 3{8}2 | 72 | 48 | 3{} | same as | |
3{8/3}2 | starry polygon, same as | ||||||||
2(144)3 | 2{8}3 | 48 | 72 | {} | |||||
2{8/3}3 | starry polygon | ||||||||
G9 4[6]2 | 192 | 24 | 4(192)2 | 4{6}2 | 96 | 48 | 4{} | same as | |
2(192)4 | 2{6}4 | 48 | 96 | {} | |||||
4{3}2 | 96 | 48 | {} | starry polygon | |||||
2{3}4 | 48 | 96 | {} | starry polygon | |||||
G10 4[4]3 | 288 | 24 | 4(288)3 | 4{4}3 | 96 | 72 | 4{} | ||
12 | 4{8/3}3 | starry polygon | |||||||
24 | 3(288)4 | 3{4}4 | 72 | 96 | 3{} | ||||
12 | 3{8/3}4 | starry polygon | |||||||
G20 3[5]3 | 360 | 30 | 3(360)3 | 3{5}3 | 120 | 120 | 3{} | self-dual, same as representation as {3,3,5} | |
3{5/2}3 | self-dual, starry polygon | ||||||||
G16 5[3]5 | 600 | 30 | 5(600)5 | 5{3}5 | 120 | 120 | 5{} | self-dual, same as representation as {3,3,5} | |
10 | 5{5/2}5 | self-dual, starry polygon | |||||||
G21 3[10]2 | 720 | 60 | 3(720)2 | 3{10}2 | 360 | 240 | 3{} | same as | |
3{5}2 | starry polygon | ||||||||
3{10/3}2 | starry polygon, same as | ||||||||
3{5/2}2 | starry polygon | ||||||||
2(720)3 | 2{10}3 | 240 | 360 | {} | |||||
2{5}3 | starry polygon | ||||||||
2{10/3}3 | starry polygon | ||||||||
2{5/2}3 | starry polygon | ||||||||
G17 5[6]2 | 1200 | 60 | 5(1200)2 | 5{6}2 | 600 | 240 | 5{} | same as | |
20 | 5{5}2 | starry polygon | |||||||
20 | 5{10/3}2 | starry polygon | |||||||
60 | 5{3}2 | starry polygon | |||||||
60 | 2(1200)5 | 2{6}5 | 240 | 600 | {} | ||||
20 | 2{5}5 | starry polygon | |||||||
20 | 2{10/3}5 | starry polygon | |||||||
60 | 2{3}5 | starry polygon | |||||||
G18 5[4]3 | 1800 | 60 | 5(1800)3 | 5{4}3 | 600 | 360 | 5{} | ||
15 | 5{10/3}3 | starry polygon | |||||||
30 | 5{3}3 | starry polygon | |||||||
30 | 5{5/2}3 | starry polygon | |||||||
60 | 3(1800)5 | 3{4}5 | 360 | 600 | 3{} | ||||
15 | 3{10/3}5 | starry polygon | |||||||
30 | 3{3}5 | starry polygon | |||||||
30 | 3{5/2}5 | starry polygon |
Polygons of the form p{2r}q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.
Polygons of the form 2{4}q are called generalized orthoplexes. They share vertices with the 4D q-q duopyramids, vertices connected by 2-edges.
Polygons of the form p{4}2 are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.
Polygons of the form p{r}p have equal number of vertices and edges. They are also self-dual.
3D perspective projections of complex polygons p{4}2 can show the point-edge structure of a complex polygon, while scale is not preserved.
The duals 2{4}p: are seen by adding vertices inside the edges, and adding edges in place of vertices.
A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has p vertices on the p-edges of the regular form.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a polygon is a plane figure made up of line segments connected to form a closed polygonal chain.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3, and each pair of nodes that is not connected by a branch at all represents a pair of mirrors at order-2.
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in . It has 27 vertices, 72 3{} edges, and 27 3{3}3 faces. It is self-dual.
In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in . 3{3}3 has 8 vertices, and 8 edges. It is self-dual. Every vertex is shared by 3 triangular edges. Coxeter named it a Möbius–Kantor polygon for sharing the complex configuration structure as the Möbius–Kantor configuration, (83).
In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.